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General Relativity and Quantum Cosmology

Title:Geometry of the 2+1 Black Hole

Abstract: The geometry of the spinning black holes of standard Einstein theory in 2+1
dimensions, with a negative cosmological constant and without couplings to
matter, is analyzed in detail. It is shown that the black hole arises from
identifications of points of anti-de Sitter space by a discrete subgroup of
$SO(2,2)$. The generic black hole is a smooth manifold in the metric sense. The
surface $r=0$ is not a curvature singularity but, rather, a singularity in the
causal structure. Continuing past it would introduce closed timelike lines.
However, simple examples show the regularity of the metric at $r=0$ to be
unstable: couplings to matter bring in a curvature singularity there. Kruskal
coordinates and Penrose diagrams are exhibited. Special attention is given to
the limiting cases of (i) the spinless hole of zero mass, which differs from
anti-de Sitter space and plays the role of the vacuum, and (ii) the spinning
hole of maximal angular momentum . A thorough classification of the elements of
the Lie algebra of $SO(2,2)$ is given in an Appendix.